Weber's class number problem and p--rationality in the cyclotomic Z^-extension of Q - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2020

Weber's class number problem and p--rationality in the cyclotomic Z^-extension of Q

Résumé

Let K:=Q(ℓ^n), n≥0, be the nth layer in the cyclotomic Z_ℓ-extension of Q. It is conjectured that, for all ℓ and n, K is principal (especially for ℓ=2, a conjecture due to Weber). Many studies (Ichimura--Morisawa--Nakajima--Okazaki...) go in this direction, as the Miller use of the Cohen--Lenstra--Martinet heuristics. Nevertheless, we examine in what circumstances a counterexample may be possible. For this, computations show that the p-torsion group T_K of the Galois group of the maximal abelian p-ramified pro-p-extension of K is not always trivial. This questions the relevance of the conjecture since #T_K=#C_K # R_K #W_K, where C_K is the p-class group of K, R_K its normalized p-adic regulator, #W_K=1 for p>2, #W_K=2^(# {v,v | 2}-1) for p=2; nevertheless, no counterexample has been found so far, even using the reflection theorem. When n increases, some relative components T_K^* may appear for large p. We give a method (Theorem 4.6), for testing #T_K≠1, allowing larger values of ℓ^n than those of the literature. Finally, we consider the subfields K of the composite Q^ of the Z_ℓ-extension and give programs finding again some rare cases of non-trivial class groups (Fukuda--Komatsu--Horie) due to genus theory in connection with a deep link involving R_K (Theorem 6.2) in relation with Greenberg's conjecture as initiated, via p-adic zeta-functions, by Taya. In all attempts, Jaulent's logarithmic class group Cl_K, K
Fichier principal
Vignette du fichier
Non-p-rational.pdf (449.84 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02935539 , version 1 (10-09-2020)
hal-02935539 , version 2 (30-09-2020)
hal-02935539 , version 3 (08-11-2020)

Identifiants

Citer

Georges Gras. Weber's class number problem and p--rationality in the cyclotomic Z^-extension of Q. 2020. ⟨hal-02935539v1⟩
112 Consultations
242 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More